The change in potential energy,when a body of mass $m$ is raised to a height $nR$ from the earth's surface is ($R =$ Radius of earth)

Two uniform solid spheres of equal radii $R$,but mass $M$ and $4M$ have a centre-to-centre separation $6R$,as shown in the figure. The two spheres are held fixed. $A$ projectile of mass $m$ is projected from the surface of the sphere of mass $M$ directly towards the centre of the second sphere. Obtain an expression for the minimum speed $v$ of the projectile so that it reaches the surface of the second sphere.

$A$ body is projected vertically upwards from the Earth's surface of radius $R$ with a velocity equal to $\frac{1}{3}$ of the escape velocity. The maximum height reached by the body is

Two hypothetical planets of masses $m_1$ and $m_2$ are at rest when they are at an infinite distance apart. Because of the gravitational force,they move towards each other along the line joining their centres. What is their speed when their separation is $d$? (Speed of $m_1$ is $v_1$ and that of $m_2$ is $v_2$)

If a body is projected vertically from the surface of the earth with a speed of $8000 \,ms^{-1}$, then the maximum height reached by the body is (Radius of the earth $= 6400 \,km$ and acceleration due to gravity $= 10 \,ms^{-2}$) (in $km$)

$A$ body starts from rest from a point at a distance $R_0$ from the centre of the Earth. The velocity acquired by the body when it reaches the surface of the Earth will be ($R$ represents the radius of the Earth).